Research On Fourth Order Finite Volume Method For Solving Highly Anisotropic Elliptic Equation

This paper mainly studies the highly anisotropic problem in the Tokamak plasma motion model,which is used to describe the electric field autonomously.However,the strong anisotropy of the model makes the limit model ill-posed as the anisotropy parameter tend to zero,which makes the linear equations in the discrete case tend to be singular.So far,the asymptotic-preserving method is considered to be the best way to deal with the numerical problems of the singular perturbation models.Based on the developed asymptotic-preserving method,this paper mainly studies the fourth order finite volume discrete scheme of the highly anisotropic elliptic equation,and uses it to get the numerical solution of the model.In this paper,we can obtain two equations equivalent to the original highly anisotropic model by using the duality-based asymptotic-preserving method and a two field iterative asymptotic-preserving method.Then based on the fundamental of the finite volume method,the fourth-order finite volume discrete scheme of the asymptotic equation is established.The error of the original and asymptotic models is also analyzed in the fourth-order finite volume framework.By analyzing the error of the numerical scheme about the original highly anisotropic model,we prove that the numerical scheme of the original model is of fourth order convergence under the finite volume framework.Its convergence is affected by the anisotropy parameter.The test results of two numerical examples show that the numerical scheme of the highly anisotropic model is of fourth order convergence when the anisotropy parameter is less than 1 but relatively close to 1.It is no longer of fourth order convergence as the anisotropy parameter is much smaller than 1.Therefore,we proves that the fourth-order convergence of the original equation's numerical scheme is related to the anisotropy parameter.By analyzing the error of the numerical scheme about the asymptotic equation,we prove that the numerical scheme of the asymptotic equation is of fourth order convergence,which is independent of anisotropy parameter.And the calculation results of the numerical examples show that the numerical scheme is of fourth order convergence regardless of the value of the asymptotic parameter,which ensures the accuracy of the conclusions numerically.It has important practical significance studying the fourth order finite volume method for highly anisotropic elliptic equations in this paper.In the study of the controlled fusion,we require that the numerical solution of the highly anisotropic elliptic equation is very accurate.However,the numerical mesh needsto be very finely divided if we use a low order numerical method,which will take up great memory of computer and consume longer computing time.We can only establish a coarse mesh to obtain the same accurate numerical solution if we use a higher order method,achieving the same precision as the low order method.In this case,the computational cost will be saved.In the next step,we can apply this results to the numerical simulation of the actual controlled fusion.